\u5b9a\u4e49\u4ee5\u4e0b\u4e24\u4e2a\u591a\u9879\u5f0f\uff1a $$ f(x) = a_0 + a_1 x + a_2 x^2 + \\dots + a_{n-1} x^{n-1} \\\\ g(x) = b_0 + b_1 x + b_2 x^2 + \\dots + b_{m-1} x^{m-1} $$ \u60f3\u8981\u6c42 $(f \\cdot g)(x)$ \u9996\u5148\u6211\u4eec\u4f1a\u60f3\u5230\u7684\u662f\u6700\u66b4\u529b\u7684\u505a\u6cd5\uff1a $$ (f \\cdot g)(x) = \\sum^{n-1}_{i=0} \\sum^{m-1}_{j=0} a_i b_j x^{i + j} $$ \u5f88\u663e\u7136\u8fd9\u79cd\u65b9\u6cd5\u975e\u5e38\u76f4\u89c2\u4e14\u5bb9\u6613\u5b9e\u73b0\uff0c\u4f46\u662f\u7f3a\u70b9\u4e5f\u5f88\u660e\u663e\uff0c\u8fd9\u4e2a\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a $O(nm)$ \uff0c\u5f53 $nm \\geq {10}^8$ \u65f6\uff0c\u5373\u4f7f\u662f\u76ee\u524d\u6700\u5f3a\u7684\u8ba1\u7b97\u673a\u4e5f\u65e0\u6cd5\u5feb\u901f\u5730\u5f97\u51fa\u7b54\u6848\uff0c\u800c\u8fd9\u79cd\u60c5\u51b5\u4e0b\u4e24\u4e2a\u591a\u9879\u5f0f\u7684\u5e73\u5747\u957f\u5ea6\u4e5f\u624d ${10}^4$ \u7684\u6570\u91cf\u7ea7\u3002\u8fd9\u663e\u7136\u4e0d\u80fd\u6ee1\u8db3\u5f88\u591a\u60c5\u51b5\u7684\u9700\u6c42\uff0c\u56e0\u4e3a\u8fd9\u4ec5\u4ec5\u53ea\u662f\u4e00\u6b21\u4e58\u6cd5\u8fd0\u7b97\uff0c\u800c\u771f\u6b63\u7684\u5e94\u7528\u573a\u666f\u4e0d\u53ef\u80fd\u53ea\u6709\u7b80\u5355\u7684\u51e0\u4e2a\u4e58\u6cd5\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u5feb\u7684\u65b9\u6cd5\u6765\u5e2e\u52a9\u6211\u4eec\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002<\/p>\n\n\n\n
\u5bf9\u4e8e\u4e00\u4e2a\u591a\u9879\u5f0f\n$$\nf(x) = a_0 + a_1 x + a_2 x^2 + \\dots + a_{n-1} x^{n-1}\n$$\n\u6211\u4eec\u663e\u7136\u77e5\u9053\u5b83\u6709 $n$ \u4e2a\u7cfb\u6570 ${ a_0, a_1, a_2, \\dots, a_{n-1} }$ \uff0c\u6709\u8fd9 $n$ \u4e2a\u7cfb\u6570\u5c31\u53ef\u4ee5\u552f\u4e00\u786e\u5b9a\u8fd9\u4e2a\u591a\u9879\u5f0f\uff0c\u90a3\u4e48\u8fd9\u65f6\u6211\u4eec\u5c31\u6765\u56de\u987e\u4e00\u4e0b\u5982\u4f55\u786e\u5b9a\u4e8c\u6b21\u51fd\u6570\u7684\u89e3\u6790\u5f0f\uff1a <\/p>\n\n\n\n
\u5bf9\u4e8e\u4e00\u4e2a\u4e8c\u6b21\u51fd\u6570: $f(x) = a x^2 + b x + c$ \uff0c\u521d\u4e2d\u6211\u4eec\u5c31\u5b66\u8fc7\u4e86\u53ef\u4ee5\u5229\u7528\u5f85\u5b9a\u7cfb\u6570\u6cd5\u6c42\u51fa $a, b, c$ \u7684\u503c\uff0c\u5373\u5229\u7528\u4e09\u4e2a\u5728\u51fd\u6570\u56fe\u50cf\u4e0a\u4e14 $y$ \u4e0d\u76f8\u7b49\u7684\u70b9\u4ee3\u5165\u5c31\u80fd\u5f97\u5230\u4e09\u5143\u4e8c\u6b21\u65b9\u7a0b\u7ec4\uff1a $$ \\left\\{ \\begin{aligned} a x^2_1 + b x_1 + c = y_1 \\\\ a x^2_2 + b x_2 + c = y_2 \\\\ a x^2_3 + b x_3 + c = y_3 \\end{aligned} \\right. \uff0c y_1 \\neq y_2 \\neq y_3 $$ \u663e\u7136\u8fd9\u4e2a\u65b9\u7a0b\u7ec4\u5b58\u5728\u552f\u4e00\u7684\u4e00\u7ec4\u89e3\uff0c\u53ea\u8981\u5bf9\u8fd9\u4e2a\u65b9\u7a0b\u7ec4\u8fdb\u884c\u6c42\u89e3\u6211\u4eec\u5c31\u53ef\u4ee5\u552f\u4e00\u786e\u5b9a\u8fd9\u4e2a\u591a\u9879\u5f0f\u3002<\/p>\n\n\n\n
\u5bf9\u4e8e $3$ \u9879\u7684\u591a\u9879\u5f0f\u6211\u4eec\u53ef\u4ee5\u8fd9\u4e48\u505a\uff0c\u663e\u7136\u5bf9\u4e8e\u6709 $n$ \u9879\u7684\u591a\u9879\u5f0f\u6211\u4eec\u4e5f\u53ef\u4ee5\u8fd9\u4e48\u505a\u3002
\u627e\u51fa $n$ \u4e2a\u4e0d\u540c\u7684\u70b9 $(x_i, f(x_i))$ \u4ee3\u5165\u5373\u53ef\u552f\u4e00\u786e\u5b9a\u8fd9\u4e2a\u591a\u9879\u5f0f\uff0c\u5373\uff1a $$ \\left\\{ \\begin{aligned} a_0 + a_1 x_0 + a_2 x^2_0 + \\dots + a_{n-1} x^{n-1}_0 & = f(x_0) \\\\
a_0 + a_1 x_1 + a_2 x^2_1 + \\dots + a_{n-1} x^{n-1}_1 & = f(x_1) \\\\
a_0 + a_1 x_2 + a_2 x^2_2 + \\dots + a_{n-1} x^{n-1}_2 & = f(x_2) \\\\
\\vdots \\\\
a_0 + a_1 x_{n-1} + a_2 x^2_{n-1} + \\dots + a_{n-1} x^{n-1}_{n-1} & = f(x_{n-1}) \\end{aligned} \\right. $$ \u540c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u8fd9\u4e2a\u65b9\u6cd5\u6c42\u51fa $(f \\cdot g)(x)$\uff0c\u800c\u663e\u7136 $(f \\cdot g)(x_i) = f(x_i) \\cdot g(x_i)$ \uff0c\u6b64\u65f6\u6211\u4eec\u5c31\u53ef\u4ee5\u7528 $O(1)$ \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u4ece\u800c\u8fbe\u5230\u7ebf\u6027\u5904\u7406\u95ee\u9898\u7684\u6548\u679c\u3002<\/p>\n\n\n\n
\u77e5\u9053\u4e86\u5b58\u5728\u7ebf\u6027\u7684\u5904\u7406\u65b9\u6cd5\uff0c\u63a5\u4e0b\u6765\u5c31\u8be5\u8003\u8651\u5982\u4f55\u8ba9\u591a\u9879\u5f0f\u7684\u8868\u793a\u5728\u4e24\u79cd\u5f62\u5f0f\u4e4b\u95f4\u8f6c\u5316\uff0c\u540c\u65f6\u8003\u8651\u8981\u627e\u7684\u70b9\u503c\u5bf9 $(x_i, f(x_i))$ \uff0c\u5982\u4f55\u627e\u5230\u5408\u9002\u7684 $x_i$ \u65b9\u4fbf\u6211\u4eec\u8ba1\u7b97\u5462\uff1f<\/p>\n\n\n\n
\u8fd9\u91cc\u5148\u63d2\u5165\u4e00\u70b9\u5c0f\u77e5\u8bc6\u3002
\u8bbe $\\omega^n_n = 1$\uff0c \u5219\u8fd9\u91cc\u7684 $\\omega_n = cos(\\frac{2 \\pi}{n}) + i sin(\\frac{2 \\pi}{n})$ \u3002
\u7531\u6b27\u62c9\u516c\u5f0f $$ e^{ix} = cos(x) + isin(x) $$ \u6240\u4ee5 $\\omega^k_n = cos(\\frac{2k \\pi}{n}) + i sin(\\frac{2k \\pi}{n})$ \uff0c\u7279\u522b\u5730\uff1a$\\omega^0_n = \\omega^n_n = 1$
\u7ecf\u8fc7\u7b80\u5355\u63a8\u5bfc\u5c31\u53ef\u4ee5\u5f97\u51fa\u4e0b\u9762\u4e09\u4e2a\u7ed3\u8bba\uff1a $$ \\begin{aligned} \\omega^{rk}_{rn} & = cos(\\frac{2rk \\pi}{rn}) + i sin(\\frac{2rk \\pi}{rn}) \\\\ & = cos(\\frac{2k \\pi}{n}) + i sin(\\frac{2k \\pi}{n}) \\\\ & = \\omega^k_n \\end{aligned} , r \\in {\\mathbb{N}}^+ , k \\in {\\mathbb{N}}^+ \\\\ \\begin{aligned} \\omega^{k+\\frac{n}{2}}_n & = cos((k + \\frac{n}{2}) \\frac{2 \\pi}{n}) + i sin((k + \\frac{n}{2}) \\frac{2 \\pi}{n}) \\\\ & = cos(\\frac{2k \\pi}{n} + \\pi) + i sin(\\frac{2k \\pi}{n} + \\pi) \\\\ & = - cos(\\frac{2k \\pi}{n}) - i sin(\\frac{2k \\pi}{n}) \\\\ & = - (cos(\\frac{2k \\pi}{n}) + i sin(\\frac{2k \\pi}{n})) \\\\ & = - \\omega^k_n \\end{aligned} , k \\in {\\mathbb{N}}^+ \\\\ \\begin{aligned} \\bar{\\omega^k_n} & = cos(\\frac{2k \\pi}{n}) - i sin(\\frac{2k \\pi}{n})\\\\ & = cos(2 \\pi - \\frac{2k \\pi}{n}) + i sin(2 \\pi - \\frac{2k \\pi}{n})\\\\ & = cos((n - k) \\frac{2 \\pi}{n}) + i sin((n - k) \\frac{2 \\pi}{n}) \\\\ & = \\omega^{n-k}_n \\end{aligned} , k \\in {\\mathbb{N}}^+ $$<\/p>\n\n\n\n
\u5bf9\u4e8e\u591a\u9879\u5f0f\uff08 $n$ \u4e3a\u5076\u6570\uff09 $$ f(x) = a_0 + a_1 x + a_2 x^2 + \\dots + a_{n - 1} x^{n - 1} $$ \u6211\u4eec\u6309\u7167\u6307\u6570\u7684\u5947\u5076\u6027\u5bf9\u5176\u5206\u7ec4 $$ \\begin{aligned} f(x) & = a_0 + a_2 x^2 \\dots + a_{n - 2} x^{n - 2} + a_1 x + a_3 x^3 \\dots + a_{n - 1} x^{n - 1} \\\\ & = a_0 + a_2 x^2 \\dots + a_{n - 2} x^{n - 2} + x (a_1 + a_3 x^2 \\dots + a_{n - 1} x^{n - 2}) \\end{aligned} $$ \u8bbe $$ \\begin{aligned} f_1(x) & = a_0 + a_ x \\dots + a_{n-2} x^{\\frac{n}{2} - 1} \\\\ f_2(x) & = a_1 + a_3 x \\dots + a_{n-1} x^{\\frac{n}{2} - 1} \\end{aligned} $$ \u6709 $$ f(x)=f_1(x^2) + x f_2(x^2) $$<\/p>\n\n\n\n
\u5f53 $x = \\omega^k_{2n}, k < n, k \\in \\mathbb{N}$ \u65f6 $$ \\begin{aligned} f(\\omega^k_{2n}) & = f_1(\\omega^{2k}_{2n}) + \\omega^k_{2n} f_2(\\omega^{2k}_{2n}) \\\\
& = f_1(\\omega^k_n) + \\omega^{\\frac{k}{2}}{n} f_2(\\omega^k_n) \\end{aligned} \\tag{1} $$ \u5f53 $x = \\omega^{k + n}_{2n}, k < n, k \\in \\mathbb{N}$ \u65f6
$$
\\begin{aligned}
f(\\omega^{k + n}_{2n}) & = f_1(\\omega^{k + n}_n) + \\omega^{k + n}_{2n} f_2(\\omega^{k + n}_n) \\\\
& = f_1(\\omega^k_n \\cdot \\omega^n_n) - \\omega^k_{2n} f_2(\\omega^k_n \\cdot \\omega^n_n) \\\\ & = f_1(\\omega^k_n) - \\omega^k_{2n} f_2(\\omega^k_n) \\ \\end{aligned} \\tag{2} $$ \u89c2\u5bdf $(1)(2)$ \u4e24\u5f0f\uff0c\u6211\u4eec\u53d1\u73b0\u77e5\u9053\u8ba1\u7b97\u51fa $f_1(\\omega^k_n), f_2(\\omega^k_n)$\uff0c\u5c31\u80fd\u8ba1\u7b97\u51fa $f(\\omega^k_{2n}), f(\\omega^{k + n}_{2n})$\uff0c\u800c\u6b64\u65f6\u7684 $f_1(\\omega^k_n), f_2(\\omega^k_n)$ \u4e5f\u53ef\u4ee5\u901a\u8fc7\u7c7b\u4f3c\u7684\u65b9\u6cd5\u8ba1\u7b97\u51fa\u6765\uff0c\u76f4\u5230
$$
f_1(\\omega^k_1) = f_2(\\omega^k_1) =1
$$
\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728 $O(log n)$ \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8ba1\u7b97\u51fa $f(\\omega^k_{2n})$ \uff0c\u4ece\u800c\u5728 $O(n log n)$ \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8ba1\u7b97\u51fa\u6211\u4eec\u8981\u7684\u6240\u6709 $f(\\omega^i_n)$ \u3002<\/p>\n\n\n\n
\u77e5\u9053\u4e86\u5982\u4f55\u8fdb\u884c\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u8003\u8651\u5982\u4f55\u5b9e\u73b0\u5b83\u7684\u9006\u53d8\u6362\u3002 <\/p>\n\n\n\n
\u4e3a\u4e86\u65b9\u4fbf\u8868\u793a\uff0c\u4e0d\u59a8\u8bbe $y_i = f(x_i)$
\u7a0d\u5fae\u6574\u7406\u4e00\u4e0b\uff1a \u6211\u4eec\u73b0\u5728\u8981\u6c42\u89e3\u7684\u662f\u4e0b\u9762\u7684\u65b9\u7a0b\u7ec4 $$ \\left\\{ \\begin{aligned} a_0 + a_1 x_0 + a_2 x^2_0 + \\dots + a_{n-1} x^{n-1}_0 & = f(x_0) \\\\
a_0 + a_1 x_1 + a_2 x^2_1 + \\dots + a_{n-1} x^{n-1}_1 & = f(x_1) \\\\
a_0 + a_1 x_2 + a_2 x^2_2 + \\dots + a_{n-1} x^{n-1}_2 & = f(x_2) \\\\
\\vdots \\\\
a_0 + a_1 x_{n-1} + a_2 x^2_{n-1} + \\dots + a_{n-1} x^{n-1}_{n-1} & = f(x_{n-1}) \\end{aligned} \\right. $$<\/p>\n\n\n\n
\u8981\u4f7f $y_i = f(x_i)$ \u65b9\u4fbf\u8ba1\u7b97\u4e14\u6ee1\u8db3\u4e24\u4e24\u4e0d\u76f8\u7b49\uff0c\u6211\u4eec\u53ef\u4ee5\u4ee4 $x_i = \\omega^i_n$ \uff0c\u5373 $$ y_i = a_0 + a_1 \\omega^i_n + a_2 (\\omega^i_n)^2 + \\dots + a_{n-1} (\\omega^i_n)^{n-1} $$ \u6211\u4eec\u4e0d\u59a8\u628a\u4e0a\u9762\u7684\u65b9\u7a0b\u7ec4\u5199\u6210\u77e9\u9635\u7684\u5f62\u5f0f\uff1a $$ \\left( \\begin{matrix} 1 & 1 & 1 & \\dots & 1 \\\\ 1 & (\\omega^1_n)^1 & (\\omega^1_n)^2 & \\dots & (\\omega^1_n)^{n-1} \\\\ 1 & (\\omega^2_n)^1 & (\\omega^2_n)^2 & \\dots & (\\omega^2_n)^{n-1} \\\\ & \\vdots & & \\ddots & \\vdots \\\\ 1 & (\\omega^{n-1}_n)^1 & (\\omega^{n-1}_n)^2 & \\dots & (\\omega^{n-1}n)^{n-1} \\
\\end{matrix}
\\right)
\\left(
\\begin{matrix}
a_0 \\\\
a_1 \\\\
a_2 \\\\
\\vdots \\\\
a_{n-1}
\\end{matrix}
\\right)
=
\\left(
\\begin{matrix}
y_0 \\\\
y_1 \\\\
y_2 \\\\
\\vdots \\\\
y_{n-1} \\end{matrix} \\right) $$ \u8bbe\u5de6\u8fb9\u7684\u65b9\u9635\u5c31\u662f\u5bf9\u5e94\u7684 $X$ \uff0c\u663e\u7136 $X$ \u53ef\u9006\uff0c\u63a5\u4e0b\u6765\u5c31\u662f\u8981\u8003\u8651\u5982\u4f55\u6c42 $X^{-1}$ \uff0c\u77e5\u9053\u4e86 $X^{-1}$ \u6211\u4eec\u5c31\u53ef\u4ee5\u8fdb\u884c\u5085\u91cc\u53f6\u9006\u53d8\u6362\u3002<\/p>\n\n\n\n
\u8fd9\u91cc\u518d\u5f15\u5165\u4e00\u4e2a\u516c\u5f0f\n$$\nx^n - 1 = (x - 1) (1 + x + x^2 + \\dots + x^{n-1})\n$$\n\u5f53 $x^n - 1 = 0$ \u65f6\uff0c\u663e\u7136 $x \\neq 0$ \uff0c\u6240\u4ee5 $1 + x + x^2 + \\dots + x^{n-1} = 0$ \uff1b
\u5f53 $x = 1$ \u65f6\uff0c$1 + x + x^2 + \\dots + x^{n-1} = n$ \uff0c\u4e14\u6ee1\u8db3\u7b49\u5f0f\u3002<\/p>\n\n\n\n
\u518d\u56de\u5230\u6c42 $X^{-1}$ \u3002
\u5bf9\u4e8e $X$ \u7684\u7b2c $i$ \u884c $$ \\left( \\begin{matrix} 1 & (\\omega^i_n) & (\\omega^i_n)^2 & \\dots & (\\omega^i_n)^{n-1} \\end{matrix} \\right) $$ \u8bbe $X^{-1}$ \u7684\u7b2c $j$ \u5217 $$ \\left( \\begin{matrix} b_0 \\\\ b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_{n-1} \\end{matrix} \\right) $$ \u5f53 $i \\neq j$ \u65f6\uff0c\u6211\u4eec\u8981\u60f3\u529e\u6cd5\u8ba9\u4e0b\u9762\u7684\u8868\u8fbe\u5f0f\u7b49\u4e8e $0$ $$ b_0 + b_1 (\\omega^i_n)^1 + b_2 (\\omega^i_n)^2 + \\dots + b_{n-1} (\\omega^i_n)^{n-1} $$ \u8fd9\u65f6\u6211\u4eec\u518d\u56de\u987e\u4e0a\u9762\u516c\u5f0f\u7684\u7b2c\u4e00\u79cd\u7ed3\u8bba\uff0c\u6709 $x = \\omega^i_n$ \uff0c\u7b49\u5f0f\u4f9d\u65e7\u6210\u7acb\u3002\u6240\u4ee5\u6211\u4eec\u5c31\u53ef\u4ee5\u5f80\u8fd9\u4e2a\u65b9\u5411\u53bb\u60f3\uff0c\u4ee4 $X^{-1}$ \u7684\u7b2c $j$ \u884c\u4e3a $$ \\left( \\begin{matrix} 1 \\\\ (\\omega^{n-j}_n) \\\\ (\\omega^{n-j}_n)^2 \\\\
\\vdots \\\\
(\\omega^{n-j}_n)^{n-1}
\\end{matrix}
\\right)
$$
\u5982\u6b64\uff0c $X^{-1}$ \u7684\u7b2c $v_{i j}$ \u4e2a\u5143\u7d20\u53ef\u4ee5\u8868\u793a\u4e3a
$$
\\begin{aligned}
v_{i j} & = 1 + \\omega^{n-j+i}_n + (\\omega^{n-j+i}_n)^2 + \\dots + (\\omega^{n-j+i}_n)^{n-1} \\\\ & = \\left\\{ \\begin{aligned} 0 , i \\neq j \\\\ n , i = j \\end{aligned} \\right. \\end{aligned} $$ \u5229\u7528\u8fd9\u4e2a\u6027\u8d28\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6784\u9020\u5c06\u65b9\u9635 $XX^{-1}$ \u7684\u4e3b\u5bf9\u89d2\u7ebf\u63a7\u5236\u4e3a $n$ \uff0c\u800c\u5176\u4ed6\u5143\u7d20\u63a7\u5236\u4e3a $0$ \uff0c\u4f8b\u5982\u4e0b\u9762\u7684\u7b2c\u4e8c\u4e2a\u65b9\u9635\u3002 $$ \\left( \\begin{matrix} 1 & 1 & 1 & \\dots & 1 \\\\ 1 & (\\omega^1_n)^1 & (\\omega^1_n)^2 & \\dots & (\\omega^1_n)^{n-1} \\\\ 1 & (\\omega^2_n)^1 & (\\omega^2_n)^2 & \\dots & (\\omega^2_n)^{n-1} \\\\ & \\vdots & & \\ddots & \\vdots \\\\ 1 & (\\omega^{n-1}_n)^1 & (\\omega^{n-1}_n)^2 & \\dots & (\\omega^{n-1}_n)^{n-1} \\end{matrix} \\right) \\left( \\begin{matrix} 1 & 1 & 1 & \\dots & 1 \\\\ 1 & (\\omega^1_n)^{n-1} & (\\omega^2_n)^{n-1} & \\dots & (\\omega^{n-1}_n)^{n-1} \\\\ 1 & (\\omega^1_n)^{n-2} & (\\omega^2_n)^{n-2} & \\dots & (\\omega^{n-1}_n)^{n-2} \\\\ & \\vdots & & \\ddots & \\vdots \\\\ 1 & (\\omega^1_n)^1 & (\\omega^2_n)^1 & \\dots & (\\omega^{n-1}_n)^{n-1} \\end{matrix} \\right) = nE $$ \u53f3\u8fb9\u7684\u77e9\u9635\u5c31\u662f\u6211\u4eec\u8981\u6c42\u7684 $nX^{-1}$ \uff0c\u518d\u901a\u8fc7\u590d\u6570\u5355\u4f4d\u6839\u7684\u7b2c\u4e09\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u5f97\u5230 $$ \\begin{aligned} X^{-1} & = \\frac{1}{n} \\left( \\begin{matrix} 1 & 1 & 1 & \\dots & 1 \\\\ 1 & (\\omega^{n-1}_n)^1 & (\\omega^{n-1}_n)^2 & \\dots & (\\omega^{n-1}_n)^{n-1} \\\\ 1 & (\\omega^{n-2}_n)^1 & (\\omega^{n-2}_n)^2 & \\dots & (\\omega^{n-2}_n)^{n-1} \\\\ & \\vdots & & \\ddots & \\vdots \\\\ 1 & (\\omega^1_n)^1 & (\\omega^1_n)^2 & \\dots & (\\omega^{n-1}_n)^{n-1} \\end{matrix} \\right) \\\\ & = \\frac{1}{n} \\left( \\begin{matrix} 1 & 1 & 1 & \\dots & 1 \\\\ 1 & \\bar{(\\omega^1_n)}^1 & \\bar{(\\omega^1_n)}^2 & \\dots & \\bar{(\\omega^1_n)}^{n-1} \\\\ 1 & \\bar{(\\omega^2_n)}^1 & \\bar{(\\omega^2_n)}^2 & \\dots & \\bar{(\\omega^2_n)}^{n-1} \\\\ & \\vdots & & \\ddots & \\vdots \\\\ 1 & \\bar{(\\omega^{n-1}_n)}^1 & \\bar{(\\omega^{n-1}_n)}^2 & \\dots & \\bar{(\\omega^{n-1}_n)}^{n-1} \\end{matrix} \\right) \\end{aligned} $$ \u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5b9e\u73b0\u4e86\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u7684\u9006\u53d8\u6362\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u2014\u2014\u52a0\u901f\u591a\u9879\u5f0f\u4e58\u6cd5\u7684\u795e\u5947\u9b54\u6cd5 0 \u4e3a\u4ec0\u4e48\u8981\u7528\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff08FFT\uff09 \u5b9a\u4e49\u4ee5\u4e0b\u4e24\u4e2a\u591a\u9879\u5f0f\uff1a $$ f(x) = a_ …<\/p>\n","protected":false},"author":1,"featured_media":133,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","emotion":"","emotion_color":"","title_style":"","license":"","footnotes":""},"categories":[1],"tags":[6],"class_list":["post-105","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-6"],"_links":{"self":[{"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/posts\/105","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/comments?post=105"}],"version-history":[{"count":16,"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/posts\/105\/revisions"}],"predecessor-version":[{"id":135,"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/posts\/105\/revisions\/135"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/media\/133"}],"wp:attachment":[{"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/media?parent=105"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/categories?post=105"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.c6-play.site\/index.php\/wp-json\/wp\/v2\/tags?post=105"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}